1/(x-22)=15+7x

Solution for 1/(x-22)=15+7x equation:

1/(x-22)=15+7x

We move all terms to the left:

1/(x-22)-(15+7x)=0


Domain of the equation: (x-22)!=0

We move all terms containing x to the left, all other terms to the right

x!=22

x∈R


We add all the numbers together, and all the variables

1/(x-22)-(7x+15)=0

We get rid of parentheses

1/(x-22)-7x-15=0

We multiply all the terms by the denominator


-7x*(x-22)-15*(x-22)+1=0

We multiply parentheses

-7x^2+154x-15x+330+1=0

We add all the numbers together, and all the variables

-7x^2+139x+331=0

a = -7; b = 139; c = +331;
Δ = b2-4ac
Δ = 1392-4·(-7)·331
Δ = 28589
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(139)-\sqrt{28589}}{2*-7}=\frac{-139-\sqrt{28589}}{-14}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(139)+\sqrt{28589}}{2*-7}=\frac{-139+\sqrt{28589}}{-14}
$